Surface acoustic wave resonator and surface acoustic wave oscillator

ABSTRACT

A surface acoustic wave resonator includes: an IDT which is disposed on a quartz crystal substrate with an Euler angle of (−1.5°≰φ≰1.5°, 117°≰θ≰142°, 41.9°≰|ψ|≰49.57°) and which excites a surface acoustic wave in an upper mode of a stop band; and an inter-electrode-finger groove formed by recessing the quartz crystal substrate between electrode fingers of the IDT, wherein the following expression: 0.01λ≰G where λ represents a wavelength of the surface acoustic wave and G represents a depth of the inter-electrode-finger groove, is satisfied and when a line occupancy of the IDT is η, the depth of the inter-electrode-finger groove G and the line occupancy η are set to satisfy the following expression: −2.5×G/λ+0.675≰η≰−2.5×G/λ+0.775.

BACKGROUND

1. Technical Field

The present invention relates to a surface acoustic wave resonator and asurface acoustic wave oscillator having the surface acoustic waveresonator, and more particularly, to a surface acoustic wave resonatorin which grooves are formed in a substrate surface and a surfaceacoustic wave oscillator having the surface acoustic wave resonator.

2. Related Art

In a surface acoustic wave (SAW) device (such as a SAW resonator), avariation in frequency-temperature characteristic is greatly affected bya stop band of the SAW or a cut angle of a quartz crystal substrate, anda shape of an IDT (Interdigital Transducer).

For example, JP-A-11-214958 discloses a configuration for exciting anupper mode and a lower mode of a stop band of a SAW and a standing wavedistribution in the upper mode and the lower mode of the stop band.

JP-A-2006-148622, JP-A-2007-208871, JP-A-2007-267033, andJP-A-2002-100959 disclose that the upper mode of the stop band has afrequency-temperature characteristic more excellent than that in thelower mode of the stop band of the SAW. JP-A-2006-148622 andJP-A-2007-208871 disclose that the cut angle of the quartz crystalsubstrate is adjusted and a normalized thickness (H/λ) of an electrodeis increased to about 0.1 so as to obtain an excellentfrequency-temperature characteristic in a SAW device using Rayleighwaves.

JP-A-2007-267033 discloses that the cut angle of the quartz crystalsubstrate is adjusted and a normalized thickness (H/λ) of an electrodeis increased to about 0.045 or greater in a SAW device using Rayleighwaves.

JP-A-2002-100959 discloses that a rotational Y-cut X-propagation quartzcrystal substrate is employed and that the frequency-temperaturecharacteristic is more improved than that in the case where theresonance in the lower mode of the stop band is used, by using theresonance in the upper end of the stop band.

In a SAW device employing an ST-cut quartz crystal substrate, groovesare disposed between electrode fingers of an IDT or between conductorstrips of a reflector, which is described in JP-A-57-5418 and“Manufacturing Conditions and Characteristics of Groove-type SAW.Resonator”, Technological Research Report of the Institute ofElectronics and Communication Engineers of Japan MW82-59 (1982). The“Manufacturing Conditions and Characteristics of Groove-type SAWResonator” also discloses that the frequency-temperature characteristicvaries depending on the depth of the grooves.

Japanese Patent No. 3851336 discloses that a configuration for setting acurve representing the frequency-temperature characteristic to athree-dimensional curve is used in the SAW device employing an LST-cutquartz crystal substrate and that any substrate with a cut angle havinga temperature characteristic represented by a three-dimensional curvecould not be discovered in a SAW device employing Rayleigh waves.

As described above, there exist various factors for improving thefrequency-temperature characteristic. Particularly, in the SAW. deviceemploying the Rayleigh waves, the increase in thickness of an electrodeof an IDT is considered as a factor contributing to thefrequency-temperature characteristic. However, the applicant of theinvention experimentally discovered that the environment resistancecharacteristic such as a temporal variation characteristic or atemperature impact resistance characteristic is deteriorated byincreasing the thickness of the electrode. For the main purpose ofimprovement in the frequency-temperature characteristic, the thicknessof the electrode should be increased as described above, and it is thusdifficult to avoid the deterioration in the temporal variationcharacteristic or the temperature impact resistance characteristic. Thisis true of a Q value and it is difficult to increase the Q value withoutincreasing the thickness of the electrode.

SUMMARY

An advantage of some aspects of the invention is that it provides asurface acoustic wave resonator and a surface acoustic wave oscillatorwhich can realize an excellent frequency-temperature characteristic,with improved environment resistance, and with a high Q value.

Some aspects of the invention can solve at least a part of the problemsmentioned above and can be embodied as the following forms orapplication examples.

APPLICATION EXAMPLE 1

Application Example 1 of the invention is directed to a surface acousticwave resonator including an IDT which is disposed on a quartz crystalsubstrate with an Euler angle of (−1.5°≦φ≦1.5°, 117°≦θ≦142°,41.9°≦|ψ|≦49.5749°) and which excites a surface acoustic wave in anupper mode of a stop band, and an inter-electrode-finger groove formedby recessing the quartz crystal substrate between electrode fingers ofthe IDT, wherein the following expression (1):0.01λ≦G  (1)where λ represents a wavelength of the surface acoustic wave and Grepresents a depth of the inter-electrode-finger groove, is satisfied,and when a line occupancy of the IDT is η, the depth of theinter-electrode-finger groove G and the line occupancy η satisfy thefollowing expression (2):−2.5×G/λ+0.675≦η≦−2.5×G/λ+0.775  (2).

According to this configuration, it is possible to improve thefrequency-temperature characteristic.

APPLICATION EXAMPLE 2

Application Example 2 of the invention is directed to the surfaceacoustic wave resonator according to Application 1, wherein the depth ofthe inter-electrode-finger groove G satisfies the following expression(3):0.01×G≦0.0695λ  (3).

According to this configuration, it is possible to suppress the shift inresonance frequency among individuals to a correction range even whenthe depth of the inter-electrode-finger groove G is deviated due tomanufacturing errors.

APPLICATION EXAMPLE 3

Application Example 3 of the invention is directed to the surfaceacoustic wave resonator according to Application 1 or 2, wherein thefollowing expression (4):0<H≦0.035λ  (4)where H represents an electrode thickness of the IDT, is satisfied.

According to this configuration, it is possible to realize the excellentfrequency-temperature characteristic in an operating temperature range.According to this configuration, it is possible to suppress thedeterioration in environment resistance with an increase in electrodethickness.

APPLICATION EXAMPLE 4

Application Example 4 of the invention is directed to the surfaceacoustic wave resonator according to Application 3, wherein the lineoccupancy η satisfies the following expression (5):η=−2.533×G/λ−2.269×H/λ+0.785  (5).

By setting the η so as to satisfy the above-mentioned expression 5 inthe electrode thickness range described in Application 3, it is possibleto set a secondary temperature coefficient within ±0.01 ppm/° C.².

APPLICATION EXAMPLE 5

Application Example 5 of the invention is directed to the surfaceacoustic wave resonator according to Application or 4, wherein the sumof the depth of the inter-electrode-finger groove G and the electrodethickness H satisfies the following expression (6):0.0407λ≦G+H  (6).

By setting the sum of the depth of the inter-electrode-finger groove Gand the electrode thickness H as expressed by the above-mentionedexpression 6, it is possible to obtain a Q value higher than that of theexisting surface acoustic wave resonator.

APPLICATION EXAMPLE 6

Application Example 6 of the invention is directed to the surfaceacoustic wave resonator according to any one of Applications 1 to 5,wherein the ψ and θ satisfy the following expression (7):ψ=1.191×10⁻³×θ³−4.490×10⁻¹×θ²+5.646×10¹×θ−2.324×10³±1.0  (7).

By manufacturing a surface acoustic wave resonator using a quartzcrystal substrate cut out at the cut angle having the above-mentionedfeature, it is possible to provide a surface acoustic wave resonatorhaving an excellent frequency-temperature characteristic in a widerange.

APPLICATION EXAMPLE 7

Application Example 7 of the invention is directed to the surfaceacoustic wave resonator according to any one of Applications 1 to 6,wherein the following expression (8):fr1<ft2<fr2  (8)wherein ft2 represents a frequency of the upper mode of the stop band inthe IDT, fr1 represents a frequency of the lower mode of the stop bandin reflectors disposed to interpose the IDT therebetween in apropagation direction of the surface acoustic wave, and fr2 represents afrequency of the upper mode of the stop band in the reflectors, issatisfied.

According to this configuration, at the frequency ft2 in the upper modeof the stop band of the IDT, the reflection coefficient |Γ| of thereflector is increased and the surface acoustic wave in the upper modeof the stop band excited from the IDT is reflected to the IDT from thereflector with a high reflection coefficient. The energy trapping of thesurface acoustic wave in the upper mode of the stop band isstrengthened, thereby implementing a surface acoustic wave resonatorwith a low loss.

APPLICATION EXAMPLE 8

Application Example 8 of the invention is directed to the surfaceacoustic wave resonator according to any one of Applications 1 to 7,wherein an inter-conductor-strip groove is disposed between conductorstrips of the reflectors, and the depth of the inter-conductor-stripgroove is smaller than the depth of the inter-electrode-finger groove.

According to this configuration, it is possible to frequency-shift thestop band of the reflector to a higher band than the stop band of theIDT. Accordingly, the relation of the above-mentioned expression 8 canbe realized.

APPLICATION EXAMPLE 9

Application Example 9 of the invention is directed to a surface acousticwave oscillator including the surface acoustic wave resonator accordingto any one of Applications 1 to 8 and an IC driving the IDT.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described with reference to the accompanyingdrawings, wherein like numbers reference like elements.

FIGS. 1A, 1B, and 1C are diagrams illustrating a configuration of a SAWdevice according to an embodiment of the invention.

FIG. 2 is a diagram illustrating the relation between an upper mode anda lower mode of a stop band.

FIG. 3 is a graph illustrating the relation between the depth of aninter-electrode-finger groove and a frequency variation in an operatingtemperature range.

FIGS. 4A to 4D are graphs illustrating a difference in secondarytemperature coefficient due to a variation in line occupancy η between aresonance point in the upper mode of the stop band and a resonance pointin the lower mode of the stop band.

FIG. 5 shows graphs illustrating the relation between the line occupancyη and the secondary temperature coefficient β when the depth of theinter-electrode-finger groove is changed with an electrode thickness of0.

FIG. 6 shows a graph illustrating the relation between the depth of theinter-electrode-finger groove in which the secondary temperaturecoefficient is 0 with the electrode thickness of 0 and the lineoccupancy η.

FIG. 7 shows graphs illustrating the relation between the line occupancyη and the frequency variation ΔF when the depth of theinter-electrode-finger groove is changed at an electrode thickness of 0.

FIG. 8 is a graph illustrating the relation between the depth of aspecific inter-electrode-finger groove when the depth of theinter-electrode-finger groove is deviated by ±0.001λ and a frequencydifference generated in the SAW resonator with the deviation.

FIG. 9 shows graphs illustrating the relation between the depth of theinter-electrode-finger groove at the secondary temperature coefficientof 0 and the line occupancy η when the electrode thickness is changed.

FIG. 10 is a diagram in which the relations between η1 in which thesecondary temperature coefficient is 0 for each electrode thickness andthe depth of the inter-electrode-finger groove are arranged in a graph.

FIG. 11 is a diagram in which the relations between the depth of theinter-electrode-finger groove and the line occupancy η are approximatedto straight lines while changing the electrode thickness from H≈0 toH=0.035λ.

FIG. 12 shows graphs illustrating the relations between the lineoccupancy η and the secondary temperature coefficient β when the depthof the inter-electrode-finger groove is changed with the electrodethickness of 0.01λ.

FIG. 13 shows graphs illustrating the relations between the lineoccupancy η and the secondary temperature coefficient β when the depthof the inter-electrode-finger groove is changed with the electrodethickness of 0.015λ.

FIG. 14 shows graphs illustrating the relations between the lineoccupancy η and the secondary temperature coefficient β when the depthof the inter-electrode-finger groove is changed with the electrodethickness of 0.02λ.

FIG. 15 shows graphs illustrating the relations between the lineoccupancy η and the secondary temperature coefficient β when the depthof the inter-electrode-finger groove is changed with the electrodethickness of 0.025λ.

FIG. 16 shows graphs illustrating the relations between the lineoccupancy η and the secondary temperature coefficient β when the depthof the inter-electrode-finger groove is changed with the electrodethickness of 0.03λ.

FIG. 17 shows graphs illustrating the relations between the lineoccupancy η and the secondary temperature coefficient β when the depthof the inter-electrode-finger groove is changed with the electrodethickness of 0.035λ.

FIG. 18 shows graphs illustrating the relations between the lineoccupancy η and the frequency variation ΔF when the depth of theinter-electrode-finger groove is changed with the electrode thickness of0.01λ.

FIG. 19 shows graphs illustrating the relations between the lineoccupancy η and the frequency variation ΔF when the depth of theinter-electrode-finger groove is changed with the electrode thickness of0.015λ.

FIG. 20 shows graphs illustrating the relations between the lineoccupancy η and the frequency variation ΔF when the depth of theinter-electrode-finger groove is changed with the electrode thickness of0.02λ.

FIG. 21 shows graphs illustrating the relations between the lineoccupancy η and the frequency variation ΔF when the depth of theinter-electrode-finger groove is changed with the electrode thickness of0.025λ.

FIG. 22 shows graphs illustrating the relations between the lineoccupancy η and the frequency variation ΔF when the depth of theinter-electrode-finger groove is changed with the electrode thickness of0.03λ.

FIG. 23 shows graphs illustrating the relations between the lineoccupancy η and the frequency variation ΔF when the depth of theinter-electrode-finger groove is changed with the electrode thickness of0.035λ.

FIG. 24 shows graphs illustrating the relations between the depth of theinter-electrode-finger groove and the Euler angle ψ when the electrodethickness and the line occupancy η are determined.

FIG. 25 is a diagram in which the relations between the depth of theinter-electrode-finger groove and the Euler angle ψ when the electrodethickness is changed are arranged in a graph.

FIG. 26 is a graph illustrating the relation between the depth of theinter-electrode-finger groove and the Euler angle ψ when the secondarytemperature coefficient β is −0.01 ppm/° C.².

FIG. 27 is a graph illustrating the relation between the depth of theinter-electrode-finger groove and the Euler angle ψ when the secondarytemperature coefficient β is +0.01 ppm/° C.².

FIG. 28 is a graph illustrating the relation between the Euler angle θand the secondary temperature coefficient β when the electrode thicknessis 0.02λ and the depth of the inter-electrode-finger groove is 0.04λ.

FIG. 29 is a graph illustrating the relation between the Euler angle φand the secondary temperature coefficient β.

FIG. 30 is a graph illustrating the relation between the Euler angle θand the Euler angle ψ in which the frequency-temperature characteristicis excellent.

FIG. 31 is a diagram illustrating examples of frequency-temperaturecharacteristic data in four sample pieces under the condition that thefrequency-temperature characteristic is the best.

FIG. 32 is a graph illustrating the relation between a height differencewhich is the sum of the depth of the inter-electrode-finger groove andthe electrode thickness and a CI value.

FIG. 33 is a table illustrating examples of a equivalent circuitconstant and a static characteristic in the SAW resonator according tothe embodiment of the invention.

FIG. 34 is a diagram illustrating impedance curve data in the SAWresonator according to the embodiment of the invention.

FIG. 35 is a graph illustrating the comparison of the relation betweenthe height difference and the Q value in the SAW resonator according tothe embodiment of the invention with the relation between the heightdifference and the Q value in the existing SAW resonator.

FIG. 36 is a diagram illustrating the SAW reflection characteristics ofthe IDT and the reflector.

FIG. 37 is a graph illustrating the relation between the electrodethickness and the frequency variation in a heat cycle test.

FIGS. 38A and 38B are diagrams illustrating the configuration of a SAWoscillator according to an embodiment of the invention.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, a surface acoustic wave resonator and a surface acousticwave oscillator according to embodiments of the invention will bedescribed in detail with reference to the accompanying drawings.

First, a surface acoustic wave (SAW) resonator according to a firstembodiment of the invention will be described with reference to FIGS.1A, 1B, and 1C. FIG. 1A is a plan view of the SAW resonator, FIG. 1B isa partially enlarged sectional view, and FIG. 1C is an enlarged viewillustrating the details of FIG. 1B.

The SAW resonator 10 according to this embodiment basically includes aquartz crystal substrate 30, an IDT 12, and a reflector 20. The quartzcrystal substrate 30 has crystal axes which are expressed by an X axis(electrical axis), a Y axis (mechanical axis), and a Z axis (opticalaxis).

In this embodiment, an in-plane rotational ST-cut quartz crystalsubstrate with Euler angles of (−1°≦φ≦1°, 117°≦θ≦142°, 41.9°≦|ψ|≦49.57°)is employed as the quartz crystal substrate 30. The Euler angle will bedescribed now. A substrate with the Euler angles of (0°, 0°, 0°) is aZ-cut substrate having a main plane perpendicular to the Z axis. Here, φof the Euler angles (φ, θ, ψ) is associated with a first rotation of theZ-cut substrate, and is a first rotation angle in which a rotatingdirection about the Z axis from the +X axis to the +Y axis is a positiverotating angle. The Euler angle θ is associated with a second rotationwhich is carried out after the first rotation of the Z-cut substrate,and is a second rotation angle in which a rotating direction about the Xaxis after the first rotation from the +Y axis after the first rotationto the +Z axis is a positive rotating angle. The cut plane of apiezoelectric substrate is determined by the first rotation angle φ andthe second rotation angle θ. The Euler angle ψ is associated with athird rotation which is carried out after the second rotation of theZ-cut substrate, and is a third rotation angle in which a rotatingdirection about the Z axis after the second rotation from the +X axisafter the second rotation to the +Y axis after the second rotation is apositive rotating angle. The propagation direction of the SAW isexpressed by the third rotation angle ψ about the X axis after thesecond rotation.

The IDT 12 includes a pair of pectinate electrodes 14 a and 14 b inwhich the base end portions of plural electrode fingers 18 are connectedto each other by a bus bar 16. The electrode fingers 18 of one pectinateelectrode 14 a (or 14 b) and the electrode fingers 18 of the otherpectinate electrode 14 b (or 14 a) are alternately arranged with apredetermined gap therebetween. Here, the electrode fingers 18 arearranged in a direction perpendicular to the X′ axis in which thesurface acoustic wave is propagated. The SAW excited by the SAWresonator 10 having the above-mentioned configuration is a Rayleigh typeSAW and has a vibration displacement component in both the Z axis afterthe third rotation and the X axis after the third rotation. In this way,by deviating the propagation direction of the SAW from the X axis whichis the crystal axis of quartz crystal, it is possible to excite the SAWin the upper mode of the stop band.

The SAW in the upper mode of the stop band and the SAW in the lower modeof the stop band will be described now. In the SAWs in the upper modeand the lower mode of the stop band formed by the regular IDT 12 shownin FIG. 2 (where the electrode fingers 18 of the IDT 12 are shown inFIG. 2), the standing waves are deviated in node and antinode positionsby π/2 from each other. FIG. 2 is a diagram illustrating a standing wavedistribution in the upper mode and the lower mode of the stop band inthe regular IDT 12.

In FIG. 2, as described above, the standing wave in the lower mode ofthe stop band indicated by a solid line has a node at the centerposition of each electrode finger 18, that is, at the reflection centerposition, and the standing wave in the upper mode of the stop bandindicated by a one-dot chained line has an antinode at the reflectioncenter position.

A pair of reflectors 20 are disposed so as to interpose the IDT 12 inthe propagation direction of the SAW. Specifically, both ends of pluralconductor strips 22 disposed parallel to the electrode fingers 18 of theIDT 12 are connected to each other.

An end-reflecting SAW resonator actively using a reflected wave from anend surface in the SAW propagation direction of the quartz crystalsubstrate or a multipair IDT-type SAW resonator exciting a standing waveof a SAW using only the IDT by increasing the number of electrode fingerpairs of the IDT does not necessarily require the reflector.

The electrode films of the IDT 12 or the reflectors 20 having theabove-mentioned configuration can be formed of aluminum (Al) or alloycontaining Al as a main component. When the alloy is used as thematerial of the electrode films, metal other than Al as a main componentcan be contained by 10% or less in terms of the weight.

In the quartz crystal substrate 30 of the SAW resonator 10 having theabove-mentioned basic configuration, grooves (inter-electrode-fingergrooves) 32 are formed between the electrode fingers of the IDT 12 orthe conductor strips of the reflectors 20.

In the grooves 32 formed in the quartz crystal substrate 30, it ispreferred that the following expression (9):0.01λ≦G  (9)where the wavelength of the SAW in the upper mode of the stop band is λand the groove depth is G, is satisfied. When the upper limit of thegroove depth G is set, as can be seen from FIG. 3., it is preferred thatthe groove depth is set in the range as expressed by the followingexpression (10).0.01λ≦G≦0.094λ  (10)By setting the groove depth G to this range, the frequency variation inthe operating temperature range (−40° C. to +85° C.) can be suppressedto 25 ppm or less as a target value the details of which will bedescribed later. The groove depth G can be preferably set to satisfy thefollowing expression (11).0.01λ≦G≦0.0695λ  (11)By setting the groove depth G to this range, the shift quantity of theresonance frequency between the individual SAW resonators 10 can besuppressed to a correction range even when a production tolerance occursin the groove depth G.

The line occupancy η is a value obtained by dividing a line width L ofeach electrode finger 18 (the width of a convex portion when a quartzcrystal convex portion is formed) by a pitch λ/2 (=L+S) between theelectrode fingers 18, as shown in FIG. 1C. Therefore, the line occupancyη can be expressed by the following expression (12).η=L/(L+S)  (12)

In the SAW resonator 10 according to this embodiment, the line occupancyη can be determined in the range expressed by the following expression(13). As can be seen from the following expression (13), η can bederived by determining the depth G of the grooves 32.−2.5×G/λ+0.675≦η≦−2.5×G/λ+0.775  (13)

The thickness of the electrode film material (of the IDT 12 or thereflectors 20) in the SAW resonator 10 according to this embodiment canbe preferably in the range expressed by the following expression (14).0<H≦0.035λ  (14)

In consideration of the electrode thickness expressed by Expression(14), the line occupancy η can be calculated by the following expression(15).η=−2.533×G/λ−2.269×H/λ+0.785  (15)

As for the line occupancy η, the production tolerance of the electricalcharacteristic (particularly, the resonance frequency) increases as theelectrode thickness increases. Accordingly, a production tolerance of±0.04 or less can occur when the electrode thickness H is in the rangeexpressed by the expression (14), and a production tolerance greaterthan ±0.04 can occur when the electrode thickness is in the range ofH>0.035λ. However, When the electrode thickness H is in the rangeexpressed by the expression (14) and the tolerance of the line occupancyη is ±0.04 or less, it is possible to embody a SAW device with a smallsecondary temperature coefficient β. That is, the line occupancy η canbe extended to the range expressed by the following expression (16)which is obtained by adding the tolerance of ±0.04 to the expression(15).η=−2.533×G/λ−2.269×H/λ+0.785+±0.04  (16)

In the SAW resonator 10 according to this embodiment having theabove-mentioned configuration, when the secondary temperaturecoefficient β is within the range of ±0.01 ppm/° C.² and the operatingtemperature range is preferably set to −40° C. to +85° C., it is a goalto improve the frequency-temperature characteristic until the frequencyvariation ΔF in the operating temperature range is 25 ppm or less. Sincethe secondary temperature coefficient β is a secondary coefficient in anapproximate polynomial of a curve representing the frequency-temperaturecharacteristic of the SAW, the small absolute value of the secondarytemperature coefficient means a small frequency variation, which meansthat the frequency-temperature characteristic is excellent. Hereinafter,it is proved by simulation that the SAW device having theabove-mentioned configuration has factors for accomplishing the subjectof the invention.

In the SAW resonator whose propagation direction is the direction of thecrystal X axis using a quartz crystal substrate called an ST cut, whenthe operating temperature range is constant, the frequency variation ΔFin the operating temperature range is about 117 ppm and the secondarytemperature coefficient β is about −0.030 ppm/° C.². In the SAWresonator which is formed using an in-plane rotation ST-cut quartzcrystal substrate in which the cut angle of the quartz crystal substrateand the SAW propagation direction are expressed by Euler angles (0°,123°, 45°) and the operating temperature range is constant, thefrequency variation ΔF is about 63 ppm and the secondary temperaturecoefficient β is about −0.016 ppm/° C.².

As described above, the variation in frequency-temperaturecharacteristic of the SAW resonator 10 is affected by the line occupancyη of the electrode fingers 18 or the electrode thickness H of the IDT 12and the groove depth G. The SAW resonator 10 according to thisembodiment employs the excitation in the upper mode of the stop band.

FIGS. 4A to 4D are graphs illustrating the variation of the secondarytemperature coefficient β when the line occupancy η is varied and theSAW is propagated by the quartz crystal substrate 30. FIG. 4A shows thesecondary temperature coefficient β in the resonance in the upper modeof the stop band when the groove depth G is 0.02λ and FIG. 4B shows thesecondary temperature coefficient β in the resonance of the lower modeof the stop band when the groove depth G is 0.02λ. FIG. 4C shows thesecondary temperature coefficient β in the resonance in the upper modeof the stop band when the groove depth G is 0.04λ and FIG. 4D shows thesecondary temperature coefficient β in the resonance of the lower modeof the stop band when the groove depth G is 0.04λ. In the simulationshown in FIGS. 4A to 4D, the SAW is propagated in some way by the quartzcrystal substrate 30 not provided with an electrode film so as to reducethe factor varying the frequency-temperature characteristic. The Eulerangle (0°, 123°, ψ) is used as the cut angle of the quartz crystal 30. Avalue at which the absolute value of the secondary temperaturecoefficient β is the minimum is properly selected as the ψ.

It can be seen from FIGS. 4A to 4D that the secondary temperaturecoefficient β greatly varies in the vicinity of the line occupancy η of0.6 to 0.7 in the upper mode and the lower mode of the stop band. Bycomparing the variation of the secondary temperature coefficient β inthe lower mode of the stop band with the variation of the secondarytemperature coefficient β in the upper mode of the stop band, it ispossible to read the following. That is, the variation of the secondarytemperature coefficient β in the lower mode of the stop band is shiftedfrom a minus side to a greater minus side and thus the characteristic isdeteriorated (the absolute value of the secondary temperaturecoefficient β increases). On the contrary, the variation of thesecondary temperature coefficient β in the upper mode of the stop bandis shifted from the minus side to a plus side and thus thecharacteristic is improved (the absolute value of the secondarytemperature coefficient β decreases).

Accordingly, in order to obtain the excellent frequency-temperaturecharacteristic in the SAW device, it is preferable to use the vibrationin the upper mode of the stop band.

The inventor made a study of the relation between the line occupancy ηand the secondary temperature coefficient β when the SAW in the uppermode of the stop band is propagated in the quartz crystal substrate withvarious groove depths G.

FIG. 5 shows simulation graphs illustrating the relations between theline occupancy η and the secondary temperature coefficient β when thegroove depth G is varied from 0.01λ (1% λ) to 0.08λ (8% λ). It can beseen from FIG. 5 that a point with β=0, that is, a point where theapproximate curve representing the frequency-temperature characteristicis a cubic curve, first appears in the vicinity of the groove depth G of0.0125λ (1.25% λ). It can be also seen from FIG. 5 that there are twopoints η with β=0 (a point (η1) with β=0 on the side where η is greatand a point (η2) with β=0 on the side where η is small). It can be alsoseen from FIG. 5 that η2 is greater than η1 in the variation of the lineoccupancy η with respect to the variation of the groove depth G.

This knowledge can be understood deeper with reference to FIG. 6. FIG. 6is a graph in which η1 and η2 are plotted in which the secondarytemperature coefficient β is 0 while varying the groove depth G. It canbe seen from FIG. 6 that η1 and η2 decrease as the groove depth Gincreases, but the variation of η2 is great in the vicinity of thegroove depth of G=0.04λ to such an extent that the variation departsfrom the graph expressed in the range of 0.5λ to 0.9λ. That is,variation of the η2 is great with respect to the variation of the groovedepth G.

FIG. 7 shows graphs in which the vertical axis of FIG. 5 is changed fromthe secondary temperature coefficient β to the frequency variation ΔF.It can be seen from FIG. 7 that the frequency variation ΔF is lowered attwo points (η1 and η2) with β=0. It can be also seen from FIG. 7 thatthe frequency variation ΔF is suppressed to be small at a pointcorresponding to η1 in any graph with the changed grooved depth G out oftwo points with β=0.

According to this tendency, it is preferable for mass products in whichproduction errors can be easily caused that the line occupancy with asmall variation of the point with β=0 relative to the variation of thegroove depth G is employed, that is, that η1 is employed. FIG. 3 shows agraph illustrating the relation between the frequency variation ΔF atthe point (η1) in which the secondary temperature coefficient β is theminimum and the grooved depth G. It can be seen from FIG. 3 that thelower limit of the groove depth G in which the frequency variation ΔF isequal to or less than 25 ppm as a target value is 0.01λ and the groovedepth G is equal to or greater than the lower limit, that is, the groovedepth range is 0.01≦G.

In FIG. 3, an example where the groove depth G is equal to or greaterthan 0.08λ in simulation is also shown. In the simulation, the groovedepth G is equal to or greater than 0.01λ, the frequency variation ΔF isequal to or less than 25 ppm, and the frequency variation ΔF decreasesas the groove depth G increases. However, when the groove depth G isequal to or greater than 0.9λ, the frequency variation ΔF increasesagain. When the groove depth is greater than 0.094λ, the frequencyvariation ΔF is greater than 25 ppm.

The graph shown in FIG. 3 is the simulation result in the state wherethe electrode films such as the IDT 12 and the reflectors 20 are notformed on the quartz crystal substrate 30, but it can be thought thatthe frequency variation ΔF of the SAW resonator 10 having the electrodefilms formed thereon is smaller, and the details of which can be seenfrom FIGS. 16 to 21. Therefore, when the upper limit of the groove depthG is determined, the maximum value in the state where the electrodefilms are not formed can be set, that is, G≦0.94λ. The range of thegroove depth G suitable for accomplishing the goal can be expressed bythe following expression (17).0.01λ≦G≦0.094λ  (17)

The groove depth G in the mass production has a tolerance of maximumabout ±0.001λ. Accordingly, when the line occupancy η is constant andthe groove depth G is deviated by ±0.001λ, the frequency variation ΔF ofeach SAW resonator 10 is shown in FIG. 8. It can be seen from FIG. 8that when the groove depth G is deviated by ±0.001λ in G=0.04λ, that is,when the groove depth is in the range of 0.039λ≦G≦0.041λ, the frequencyvariation ΔF is about ±500 ppm.

Here, when the frequency variation ΔF is less than ±1000 ppm, thefrequency can be adjusted by the use of various means for finelyadjusting the frequency. However, when the frequency variation ΔF isequal to or greater than ±1000 ppm, the static characteristic such asthe Q value and the CI (Crystal Impedance) value and the long-termreliability are affected by the frequency adjustment, whereby the goodproduction rate of the SAW resonator 10 is deteriorated.

By deriving an approximate expression representing the relation betweenthe frequency variation ΔF [ppm] and the groove depth G from thestraight line connecting the plots shown in FIG. 8, the followingexpression (18) can be obtained.ΔF=16334G−137  (18)

Here, the range of G satisfying ΔF<1000 ppm is G≦0.0695λ. Therefore, therange of the groove depth G according to this embodiment is preferablyexpressed by the following expression (19).0.01λ≦G≦0.0695λ  (19)

FIG. 9 shows graphs illustrating the relation between η with thesecondary temperature coefficient of β=0, that is, the line occupancy ηrepresenting a tertiary temperature characteristic, and the groove depthG. The quartz crystal substrate 30 has the Euler angle of (0°, 123°, ψ).Here, an angle at which the frequency-temperature characteristic showsthe tendency of the cubic curve, that is, an angle at which thesecondary temperature coefficient is β=0, is properly selected as ψ. Therelations between the Euler angle ψ at which η with β=0 is obtained andthe groove depth G under the same condition as shown in FIG. 9 are shownin FIG. 24. In the graph with the electrode thickness of H=0.02λ in FIG.24, the plot of ψ<42° is not shown, but ψ=41.9° at G=0.03λ is shown inthe plot of η2 of the graph. The plot of the relation between the groovedepth G at each electrode thickness and the line occupancy η is obtainedfrom FIGS. 12 to 17 the details of which are described later.

It can be seen from FIG. 9 that the variation of η1 due to the variationof the groove depth G is smaller than the variation of η2 with anythickness, as described above. Accordingly, η1 is extracted from thegraph of thicknesses in FIG. 9 and is arranged in FIG. 10. It can beseen from FIG. 10 that η1 is concentrated in the line indicated by abroken line. In FIG. 10, the plot indicating the upper limit of the lineoccupancy η represents the SAW resonator with the electrode thickness ofH=0.01λ and the plot indicating the lower limit of the line occupancy ηrepresents the SAW resonator with the electrode thickness of H=0.035λ.That is, as the electrode thickness H increases, the line occupancy η inwhich the secondary temperature coefficient is β=0 decreases.

By calculating the approximate expression of the plot indicating theupper limit of the line occupancy η and the plot indicating the lowerlimit of the line occupancy η on the basis of the above description, thefollowing expressions (20) and (21) can be derived.η=−2.5×G/λ+0.775  (20)η=−2.5×G/λ+0.675  (21)

It can be said from the above expressions (20) and (21) that η in therange surrounded with the broken line in FIG. 10 can be determined inthe range expressed by the following expression (22).−2.5×G/λ+0.675≦η≦−2.5×G/λ+0.775  (22)

Here, when the secondary temperature coefficient β is permitted within±0.01 ppm/° C.², it is confirmed that expressions (19) and (22) are bothsatisfied and thus the secondary temperature coefficient β is in therange of ±0.01 ppm/° C.².

When the relations between the groove depth G with β=0 and the lineoccupancy η in the SAW resonators 10 with the electrode thickness ofH≈0, 0.01λ, 0.02λ, 0.03λ, and 0.035λ are expressed by the approximatestraight line on the basis of the expressions (20) to (22), the straightlines shown in FIG. 11 are obtained. The relations between the groovedepth G and the line occupancy η in the quartz crystal substrate 30 nothaving an electrode film formed thereon are as shown in FIG. 6.

The relational expression between the groove depth G and the lineoccupancy η in which the frequency-temperature characteristic isexcellent can be expressed by the following expression (23) on the basisof the approximate expressions indicating the approximate straight lineswith the electrode thicknesses H.η=−2.533×G/λ−2.269×H/λ+0.785  (23)

As for the line occupancy η, the production tolerance of the electricalcharacteristic (particularly, the resonance frequency) increases as theelectrode thickness increases. Accordingly, a production tolerance of±0.04 or less can occur when the electrode thickness H is in the rangeexpressed by expression (14), and a production tolerance greater than±0.04 can occur when the electrode thickness is in the range ofH>0.035λ. However, when the electrode thickness H is in the rangeexpressed by the expression (23) and the tolerance of the line occupancyη is ±0.04 or less, it is possible to embody a SAW device with a smallsecondary temperature coefficient β. That is, when the secondarytemperature coefficient β is set to ±0.01 ppm/° C.² or less inconsideration of the production tolerance of the line occupancy, theline occupancy η can be extended to the range expressed by the followingexpression (24) which is obtained by adding the tolerance of ±0.04 tothe expression (23).η=−2.533×G/λ−2.269×H/λ+0.785+0.04  (23)

Graphs illustrating the relations between the line occupancy η and thesecondary temperature coefficient β when the electrode thickness ischanged to 0.01λ (1% λ), 0.015λ (1.5% λ), 0.02λ (2% λ), 0.025λ (2.5% λ),0.03λ (3% λ), and 0.035λ (3.5% λ) and the groove depth G is changed areshown in FIGS. 12 to 17.

Graphs illustrating the relations between the line occupancy η and thefrequency variation ΔF in the SAW resonators 10 corresponding to FIGS.12 to 17 are shown in FIGS. 18 to 23. All the quartz crystal substrateshave the Euler angle of (0°, 123°, ψ) and an angle at which ΔF is theminimum is properly selected as ψ.

Here, FIG. 12 is a diagram illustrating the relation between the lineoccupancy η and the secondary temperature coefficient β when theelectrode thickness H is 0.01λ and FIG. 18 is a diagram illustrating therelation between the line occupancy η and the frequency variation ΔFwhen the electrode thickness H is 0.01λ.

FIG. 13 is a diagram illustrating the relation between the lineoccupancy η and the secondary temperature coefficient β when theelectrode thickness H is 0.015λ and FIG. 19 is a diagram illustratingthe relation between the line occupancy η and the frequency variation ΔFwhen the electrode thickness H is 0.015λ.

FIG. 14 is a diagram illustrating the relation between the lineoccupancy η and the secondary temperature coefficient β when theelectrode thickness H is 0.02λ and FIG. 20 is a diagram illustrating therelation between the line occupancy η and the frequency variation ΔFwhen the electrode thickness H is 0.02λ.

FIG. 15 is a diagram illustrating the relation between the lineoccupancy η and the secondary temperature coefficient β when theelectrode thickness H is 0.025λ and FIG. 21 is a diagram illustratingthe relation between the line occupancy η and the frequency variation ΔFwhen the electrode thickness H is 0.025λ.

FIG. 16 is a diagram illustrating the relation between the lineoccupancy η and the secondary temperature coefficient β when theelectrode thickness H is 0.03λ and FIG. 22 is a diagram illustrating therelation between the line occupancy η and the frequency variation ΔFwhen the electrode thickness H is 0.03λ.

FIG. 17 is a diagram illustrating the relation between the lineoccupancy η and the secondary temperature coefficient β when theelectrode thickness H is 0.035λ and FIG. 23 is a diagram illustratingthe relation between the line occupancy η and the frequency variation ΔFwhen the electrode thickness H is 0.035λ.

In the drawings (FIGS. 12 to 23), a minute difference exists in thegraphs, but it can be seen that the variation tendency is similar toFIGS. 5 and 7 which are the graphs illustrating the relations betweenthe line occupancy η and the secondary temperature coefficient β andbetween the line occupancy η and the frequency variation ΔF only in thequartz crystal substrate 30.

That is, it can be said that the advantage of this embodiment can beobtained in the propagation of the surface acoustic wave only in thequartz crystal substrate 30 excluding the electrode films.

The relations between ψ acquired from η1 in the graphs shown in FIG. 24and the groove depth G are arranged in FIG. 25. The reason for selectingη1 is as described above. As shown in FIG. 25, even when the electrodethickness is changed, it can be seen that the angle of ψ is hardlychanged and the optimal angle of ψ varies with the variation of thegroove depth G. This proves that the variation of the secondarytemperature coefficient β is greatly affected by the shape of the quartzcrystal substrate 30.

In the same way as described above, the relations of the groove depth Gto ψ at which the secondary temperature coefficient is β=−0.01 ppm/° C.²and ψ at which the secondary temperature coefficient is β=+0.01 ppm/°C.² are acquired and arranged in FIGS. 26 and 27. When the angle of ψsatisfying −0.01≦β≦+0.01 is calculated from the graphs (FIGS. 25 to 27),the angle range of ψ under the above-mentioned condition can bedetermined preferably as 43°≦ψ≦45° and more preferably as 43.2°≦ψ≦44.2°.

The variation of the secondary temperature coefficient β when the angleof η is given, that is, the relation between θ and the secondarytemperature coefficient β, is shown in FIG. 28. Here, the SAW deviceused in the simulation includes a quartz crystal substrate in which thecut angle and the SAW propagation direction are expressed by the Eulerangle (0, θ, ψ) and the groove depth G is 0.04λ, where the electrodethickness H is 0.02λ. As for ψ, a value at which the absolute value ofthe secondary temperature coefficient β is the minimum is selected inthe above-mentioned angle range on the basis of the set angle of θ. η isset to 0.6383 on the basis of the expression 23.

Under this condition, it can be seen from FIG. 28 illustrating therelation between θ and the secondary temperature coefficient β that whenθ is in the range of 117° to 142°, the absolute value of the secondarytemperature coefficient β is in the range of 0.01 ppm/° C.². Therefore,by determining θ in the range of 117°≦θ≦142° with the above-mentionedset value, it can be said that it is possible to implement the SAWresonator 10 having an excellent frequency-temperature characteristic.

FIG. 29 is a graph illustrating the relation between the angle of φ andthe secondary temperature coefficient β when the groove depth G is0.04λ, the electrode thickness H is 0.02λ, and the line occupancy η is0.65 in the quartz crystal substrate 30 with the Euler angle of (φ,123°, 43.77°).

It can be seen from FIG. 29 that the secondary temperature coefficient βis lower than −0.01 when φ is −2° and +2°, but the absolute value of thesecondary temperature coefficient β is in the range of 0.01 when φ is inthe range of −1.5° to +1.5°. Therefore, by determining φ in the range of−1.5°≦φ≦+1.5° and preferably −1°≦φ≦+1° with the above-mentioned setvalue, it is possible to implement the SAW resonator 10 with anexcellent frequency-temperature characteristic.

In the above description, the optimal ranges of φ, θ, and ψ are derivedfrom the relation to the groove depth G under a predetermined condition.On the contrary, FIG. 30 shows the very desirable relation between θ andψ which the frequency variation is the minimum in the range of −40° C.to +85° C. and the approximate expression thereof is calculated. Asshown in FIG. 30, the angle of ψ varies with the rising of the angle ofθ and rises to draw a cubic curve. In the example shown in FIG. 30, ψ is42.79° at θ=117° and ψ is 49.57° at θ=142°. The approximate curve ofthese plots is the curve indicated by the broken line in FIG. 30 and canbe expressed by the following expression (25) as an approximateexpression.ψ=1.19024×10⁻³×θ³−4.48775×10⁻¹×θ²+5.64362×10¹×θ−2.32327×10³=1.0  (25)

From this expression, ψ can be determined by determining θ and the rangeof ψ when the range of θ is set to the range of 117°≦θ≦142° can be setto 42.79°≦ψ≦49.57°. The groove depth G and the electrode thickness H inthe simulation are set to G=0.04λ and H=0.02λ.

For the above-mentioned reason, in this embodiment, by implementing theSAW resonator 10 under various predetermined conditions, it is possibleto obtain a SAW resonator with an excellent frequency-temperaturecharacteristic satisfying a target value.

In the SAW resonator 10 according to this embodiment, as shown in theexpression (14) and FIGS. 12 to 23, it is possible to improve thefrequency-temperature characteristic after the electrode thickness H isset to the range of 0<H≦035λ. Unlike the improvement of thefrequency-temperature characteristic by greatly increasing the thicknessH in the related art, it is possible to improve thefrequency-temperature characteristic while maintaining the environmentresistance characteristic. FIG. 37 shows the relation between theelectrode thickness (Al electrode thickness) and the frequency variationin a heat cycle test. The result of the heat cycle test shown in FIG. 37is obtained after the cycle that the SAW resonator is exposed to theatmosphere of −55° C. for 30 minutes and is then exposed to theatmosphere of +125° C. for 30 minutes is repeated eight times. It can beseen from FIG. 37 that the frequency variation (F variation) in therange of the electrode thickness H of the SAW resonator 10 according tothis embodiment is equal to or less than ⅓ of that in the case where theelectrode thickness H is 0.06λ and the inter-electrode-finger groove isnot disposed. In any plot of FIG. 37, H+G=0.06λ is set.

A high-temperature shelf test of leaving a sample in the atmosphere of125° C. for 1000 hours was performed on the SAW resonator produced underthe same condition as shown in FIG. 37. It was confirmed that thefrequency variation before and after the test of the SAW resonator(under four conditions of H=0.03λ and G=0.03λ, H=0.02λ and G=0.04λ,H=0.015λ and G=0.045λ, and H=0.01λ and G=0.05λ) is equal to or less than⅓ of that of the existing SAW resonator (under the condition of H=0.06λand G=0).

In the SAW resonator 10 produced under the same conditions as describedabove and the conditions of H+G=0.067λ (with an aluminum thickness 2000Å and a groove depth of 4700 Å), the line occupancy of the IDT ofηi=0.6, the line occupancy of the reflector of ηr=0.8, the Euler angleof (0°, 123°, 43.5°), the number of IDT pairs of 120, the intersectionwidth of 40λ (λ=10 μm), the number of reflectors (one side) of 72 (36pairs), and the tilt angle of the electrode fingers of zero (thearrangement direction of the electrode fingers is equal to the phasespeed direction of the SAW), the frequency-temperature characteristicshown in FIG. 31 is obtained.

FIG. 31 is a graph in which the frequency-temperature characteristics offour test samples (n=4) are plotted. It can be seen from FIG. 31 thatthe frequency variation ΔF in the operating temperature range of thetest samples is equal to or less than about 20 ppm.

In this embodiment, the influence on the frequency-temperaturecharacteristic depending on the groove depth G and the electrodethickness H has been described. However, the depth (height difference)which is the sum of the groove depth G and the electrode thickness Haffects the static characteristics such as the equivalent circuitconstant and the CI value or the Q value. For example, a graphillustrating the relation between the height difference and the CI valuewhen the height difference is changed in the range of 0.062λ to 0.071λis shown in FIG. 32. It can be seen from FIG. 32 that the CI valueconverges at the height difference of 0.067λ and is not changed (notlowered) even at a greater height difference.

The frequency, the equivalent circuit constant, and the staticcharacteristics in the SAW resonator 10 having the frequency-temperaturecharacteristic shown in FIG. 31 are arranged in FIG. 33. Here, Frepresents the frequency, Q represents the Q value, γ represents acapacity ratio, CI represents the CI (Crystal Impedance) value, and Mrepresents the performance index (figure of merit).

FIG. 35 shows a graph illustrating the comparison of the relations ofthe height difference and the Q value in the existing SAW resonator andthe SAW resonator 10 according to this embodiment. In FIG. 35, the graphindicated by the bold line represents the characteristic of the SAWresonator 10 according to this embodiment, where the grooves aredisposed between the electrode fingers and the resonance in the uppermode of the stop band is used. The graph indicated by the thin linerepresents the characteristic of the existing SAW resonator, where thegrooves are not disposed in the electrode fingers and the resonance inthe upper mode of the stop band is used. As can be clearly seen fromFIG. 35, when the grooves are disposed between the electrode fingers andthe resonance in the upper mode of the stop band is used, the Q value inthe region where the height difference (G+H) is equal to or greater than0.0407λ (4.07% λ) is higher than that in the case where the grooves arenot disposed between the electrode fingers and the resonance in thelower mode of the stop band is used.

The basic data of the SAW resonator in the simulation is as follows. Thebasic data of the SAW resonator 10 according to this embodiment includesH: 0.02λ, G: variable, IDT line occupancy ηi: 0.6, reflector lineoccupancy ηr: 0.8, Euler angle: (0°, 123°, 43.5°), number of pairs: 120,intersection width: 40λ (λ=10 μm), number of reflectors (one side): 60,and no tilt angle of electrode finger. The basic data of the existingSAW resonator includes H: variable, G: zero, IDT line occupancy ηi: 0.4,reflector line occupancy ηr: 0.3, Euler angle: (0°, 123°, 43.5°), numberof pairs: 120, intersection width: 40λ (λ=10 μm), number of reflectors(one side): 60, and no tilt angle of electrode finger.

By referring to FIG. 33 or 35 for the purpose of comparison of thecharacteristics of the SAW resonators, it can be understood how the SAWresonator 10 according to this embodiment increases in Q value. It isthought that the increase in Q value is due to the improvement of theenergy trapping effect and the reason is as follows.

In order to efficiently trap the energy of the surface acoustic waveexcited in the upper mode of the stop band, the upper end frequency ft2of the stop band of the IDT 12 can be set between the lower endfrequency fr1 of the stop band of the reflector 20 and the upper endfrequency fr2 of the stop band of the reflector 20, as shown in FIG. 36.That is, the frequencies can be set to satisfy the following expression(26).fr1<ft2<fr2  (26)

Accordingly, the reflection coefficient Γ of the reflector 20 becomesgreater at the upper end frequency ft2 of the stop band of the IDT 12and the SAW in the upper mode of the stop band excited from the IDT 12is reflected to the IDT 12 with a higher reflection coefficient by thereflector 20. The energy trapping force of the SAW in the upper mode ofthe stop band is strengthened, thereby embodying a resonator with a lowloss.

On the contrary, when the relation among the upper end frequency ft2 ofthe stop band of the IDT 12, the lower end frequency fr1 of the stopband of the reflector 20, and the upper end frequency fr2 of the stopband of the reflector 20 is set to ft2<fr1 or fr2<ft2, the reflectioncoefficient Γ of the reflector 20 at the upper end frequency ft2 of thestop band of the IDT 12 becomes smaller and it is thus difficult toobtain the strong energy trapping.

Here, in order to realize the state expressed by the expression (26), itis necessary to frequency-shift the stop band of the reflector 20 to thehigher band side than the stop band of the IDT 12. Specifically, thisstate can be realized by setting the arrangement pitch of the conductorstrips 22 of the reflector 20 to be smaller than the arrangement pitchof the electrode fingers 18 of the IDT 12. In another method, thethickness of the electrode film formed as the conductor strips 22 of thereflector 20 can be set to be smaller than the thickness of theelectrode film formed as the electrode fingers 18 of the IDT 12 or thedepth of the inter-conductor-strip groove of the reflector 20 can be setto be smaller than the depth of the inter-electrode-finger groove of theIDT 12. Two or more of the methods may be combined.

As can be clearly seen from FIG. 33, it is possible to obtain a highfigure of merit Min addition to the increase in Q value. FIG. 34 is agraph illustrating the relation between the impedance Z and thefrequency in the SAW resonator having the characteristics shown in FIG.33. It can be seen from FIG. 34 that no useless spurious exists in thevicinity of the resonance point.

In the IDT 12 of the SAW resonator 10 according to this embodiment, allthe electrode fingers are alternately intersected. However, the SAWresonator 10 according to the invention can exhibit the considerableadvantage using only the quartz crystal substrate. Accordingly, evenwhen the electrode fingers 18 of the IDT 12 are removed, the sameadvantage can be obtained.

The grooves 32 may be disposed partially between the electrode fingers18 or between the conductor strips 22 of the reflector 20. Particularly,since the center portion of the IDT 12 with a high vibrationdisplacement greatly affects the frequency-temperature characteristic,the grooves 32 may be disposed only in the center portion. With thisconfiguration, it is possible to provide the SAW resonator 10 with anexcellent frequency-temperature characteristic.

In the above-mentioned embodiment, Al or an alloy containing Al as amain component is used for the electrode films. However, another metalmay be used for the electrode films as long as it provides the sameadvantages as the above-mentioned embodiment.

Although a one-terminal-pair SAW resonator having only one IDT isexemplified in the above-mentioned embodiment, the invention can beapplied to a two-terminal-pair SAW resonator having plural IDTs and canbe also applied to a vertical-coupling or horizontal-couplingdouble-mode SAW filter or multimode SAW filter.

A SAW oscillator according to an embodiment of the invention will bedescribed with reference to FIGS. 38A and 38B. As shown in FIGS. 38A and38B, the SAW oscillator according to this embodiment includes theabove-mentioned SAW resonator 10, an IC (Integrated Circuit) 50controlling the driving of the SAW resonator by applying a voltage tothe IDT 12 of the SAW resonator 10, and a package receiving theelements. FIG. 38A is a plan view in which leads are excluded and FIG.38B is a sectional view taken along line XXXVIIIB-XXXVIIIB of FIG. 38A.

In the SAW oscillator 100 according to this embodiment, the SAWresonator 10 and the IC 50 are received in the same package 56, andelectrode patterns 54 a to 54 g formed on a bottom plate 56 a of thepackage 56, the pectinate electrodes 14 a and 14 b of the SAW resonator10, and pads 52 a to 52 f of the IC 50 are connected to each other bymetal wires 60. The cavity of the package 56 receiving the SAW resonator10 and the IC 50 are air-tightly sealed with a lid 58. According to thisconfiguration, the IDT 12 (see FIGS. 1A to 1C), the IC 50, and externalmounting electrodes (not shown) formed on the bottom surface of thepackage 56 can be electrically connected to each other.

The entire disclosure of Japanese Patent Application No. 2009-045359,filed Feb. 27, 2009, Japanese Patent Application No. 2009-050112, filedMar. 4, 2009 and Japanese Patent Application No. 2009-285224, filed Dec.16, 2009 are expressly incorporated by reference herein.

1. A surface acoustic wave resonator comprising: an IDT which isdisposed on a quartz crystal substrate with an Euler angle of(−1.5°≦φ≦1.5°, 117°≦θ≦142°, 41.9°≦|ψ|≦49.57°) and which excites asurface acoustic wave in an upper mode of a stop band; and aninter-electrode-finger groove formed by recessing the quartz crystalsubstrate between electrode fingers of the IDT, wherein the followingexpression:0.01λ≦G where λ represents a wavelength of the surface acoustic wave andG represents a depth of the inter-electrode-finger groove, is satisfiedand when a line occupancy of the IDT is η, the depth of theinter-electrode-finger groove G and the line occupancy η are set tosatisfy the following expression:−2.5×G/λ+0.675≦η≦−2.5×G/λ+0.775.
 2. The surface acoustic wave resonatoraccording to claim 1, wherein the depth of the inter-electrode-fingergroove G satisfies the following expression:0.01λ≦G≦0.0695λ.
 3. The surface acoustic wave resonator according toclaim 1, wherein the following expression:0<H≦0.035λ where H represents an electrode thickness of the IDT, issatisfied.
 4. The surface acoustic wave resonator according to claim 3,wherein the line occupancy η satisfies the following expression:η=−2.533×G/λ−2.269×H/λ+0.785+±0.04.
 5. The surface acoustic waveresonator according to claim 3, wherein the sum of the depth of theinter-electrode-finger groove G and the electrode thickness H satisfiesthe following expression:0.0407λ≦G+H.
 6. The surface acoustic wave resonator according to claim1, wherein the ψ and θ satisfy the following expression:ψ=1.191×10⁻³×θ³−4.490×10⁻¹×θ²+5.646×10¹×θ−2.324×10³±1.0.
 7. The surfaceacoustic wave resonator according to claim 1, wherein the followingexpression:fr1<ft2<fr2 where ft2 represents a frequency of the upper mode of thestop band in the IDT, fr1 represents a frequency of the lower mode ofthe stop band in reflectors disposed to interpose the IDT therebetweenin a propagation direction of the surface acoustic wave, and fr2represents a frequency of the upper mode of the stop band in thereflectors, is satisfied.
 8. The surface acoustic wave resonatoraccording to claim 1, wherein an inter-conductor-strip groove isdisposed between conductor strips of the reflectors, and wherein thedepth of the inter-conductor-strip groove is smaller than the depth ofthe inter-electrode-finger groove.
 9. A surface acoustic wave oscillatorcomprising the surface acoustic wave resonator according to claim 1 andan IC driving the IDT.